Literally, erforderlich translates to required, and notwendig translates to necessary.
In most cases, they are fully interchangeable.
erforderlich might bear some associations, that the object is "required by someone", or rather the requirement is "imposed by someone", while notwendig could imply a more objective necessity. This is due to the fact, that the stem of erforderlich is the stem of the verb fordern which translates to demand, ask for, call for, require, hence erforderlich could ring a bell of a social interaction. Thus, using erforderlich might frame the message in a more personal way.
But the nuance is very small, and it is not even a necessary association - the verb erfordern means to necessitate, to require, to need, which is closer to an objective frame, than the mere stem fordern. You can easily say sentences like
Die Situation erforderte schnelles Handeln (The situation needed quick action.)
where there is no person involved requiring anything, but erfordern expresses some rather objective necessity.
Use and interchangeability of erforderlich and notwendig are very analogous to their English translations, I'd say.
Style and Tone
From my experience I would say that notwendig, or even nötig is more common and more colloquial, and erforderlich is more bureaucratic and formal.